The dumper generally has the advantage because the feeling is already lost and has gone to the point that they have a reason to not to be with a person anymore. There were weeks, maybe months before the dumper finally decides to end a relationship while the dumpee has no idea what's coming. The dumper is a step ahead because they have already begun to move on. Then there is could be other factors such as the dumper has already moved on or at least has someone else in their sights. The dumpee feels the worst unless they already have someone else that can replace the person dumping them. Then think about this. You get dumped, and you see your ex with someone else and you haven't totally moved on or vice versa. The dumper rarely feels any type of pain unless it's really for themselves.
Dumpers feel the power of having control over someone's happiness for a moment and get a kick out of it (especially girls). They feel that they are doing the right thing. Then there is the excuse factor. Some people get dumped and get excuses told to them on why they are being dumped, leaving overall confusion. It's like when a guy is told that he is a great catch, nice, sweet, and the caring guy that girls "ask" for, but is always being dumped. Since they don't tell him the truth on how to change his game so he can get other girls in the future, he lingers on and tries to figure out why, thus the pain lasting longer while the dumper has already moved on to someone else.
It depends on the people and the situations, but it's usually the dumper that feels freedom while the dumpee has their world shattered. By the way, in many cases it is not a good idea to attempt to be "friends" after a relationship. The "friendship" ends when one has feelings and/or sexual attraction for the other.